Depth estimation from downward continuation: An entropy-based approach to normalized full gradient

The normalized full gradient is based on the noticeable stability of the modulus of the analytic signal of downward potential fields. We have developed a new approach to the study of the normalized full gradient, based on assessing the entropy of the normalized modulus of the analytic signal at each level of continuation. The increased disorder of progressively downward continued fields implies an increase of the computed entropy. However, a local decrease of the entropy is expected at the source level, where the field gets singular, as entropy decreases when the information is concentrated. Thus, our method is based on a simple search for a minimum of the computed entropy versus depth curve, and the estimated depth will be that at which the minimum is attained. The method is sensitive to interference and other types of noise, and specific strategies to deal with these limitations are defined and tested on synthetic data. The depth estimate is obtained without the assumption of a specific source shape. The depth could correspond to the top or center in case of a simple, one-point source, or it may be related to an intermediate depth between the source top and its center in case of a finite, general source. We applied this method to real magnetic data from an unexploded ordnance survey, and it could verify a rather accurate depth-to-source estimate when compared with excavation results.